Johansson (2011) in "Hail the impossible: p-values, evidence, and likelihood" (here is also link to the journal) states that lower $p$-values are often considered as stronger evidence against the null. Johansson implies that people would consider evidence against the null to be stronger if their statistical test outputted a $p$-value of $0.01$, than if their statistical test outputted a $p$-value of $0.45$. Johansson lists four reasons why the $p$-value cannot be used as evidence against the null:
- $p$ is uniformly distributed under the null hypothesis and can therefore never indicate evidence for the null.
- $p$ is conditioned solely on the null hypothesis and is therefore unsuited to quantify evidence, because evidence is always relative in the sense of being evidence for or against a hypothesis relative to another hypothesis.
- $p$ designates probability of obtaining evidence (given the null), rather than strength of evidence.
- $p$ depends on unobserved data and subjective intentions and therefore implies, given the evidential interpretation, that the evidential strength of observed data depends on things that did not happen and subjective intentions.
Unfortunately I cannot get an intuitive understanding from Johansson's article. To me a $p$-value of $0.01$ indicates there is less chance the null is true, than a $p$-value of $0.45$. Why are lower $p$-values not stronger evidence against null?